A068997 Numbers k such that Sum_{d|k} d*mu(d) divides k.

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 64, 72, 80, 84, 96, 100, 108, 120, 128, 144, 160, 162, 168, 192, 200, 216, 240, 252, 256, 272, 288, 312, 320, 324, 336, 360, 384, 400, 432, 440, 480, 486, 500, 504, 512, 544, 576, 588, 600, 624, 640, 648

Offset

1,2

Comments

Numbers k such that A023900(k) divides k.

The only squarefree terms so far are a(1), a(2), and a(4). - Torlach Rush, Dec 04 2017

There are no more squarefree terms. The squarefree terms are also the squarefree terms of A007694 since A023900(n) = A008683(n) * A000010(n) for squarefree numbers n, and A007694 contains only 3-smooth numbers (A003586). - Amiram Eldar, Apr 19 2025

There is a surjective mapping from all even numbers not in this sequence to terms of the sequence. The first such is 10 to a(9). The next is 14, 28, 42 to a(19). All even numbers not in the sequence are divisors of some term in the sequence. - Torlach Rush, Dec 08 2017

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Maple

with(numtheory): A068997 := i->`if`(i mod phi(mul(j, j=factorset(i)))=0, i, NULL): seq(A068997(i), i=1..650); # Peter Luschny, Nov 02 2010

Mathematica

Select[Range[650], Divisible[#, DivisorSum[#, # MoebiusMu[#] &]] &] (* Michael De Vlieger, Nov 20 2017 *)
q[1] =True; q[n_] := Divisible[n, Times @@ ((First[#] - 1) & /@ FactorInteger[n])]; Select[Range[650], q] (* Amiram Eldar, Apr 19 2025 *)

Prog

(PARI) for(n=1, 1000, if(n%sumdiv(n, d, moebius(d)*d)==0, print1(n, ", ")))
(PARI) isok(k) = !(k % vecprod(apply(x -> 1-x, factor(k)[, 1]))); \\ Amiram Eldar, Apr 19 2025
(Haskell)
a068997 n = a068997_list !! (n - 1)
a068997_list = filter (\x -> mod x (a173557 x) == 0) [1..]
-- Reinhard Zumkeller, Jun 01 2015

Crossrefs

Cf. A000010, A003586, A007694, A008683, A023900, A173557, A293928.

Sequence in context: A320580 A325763 A363949 * A293928 A067712 A060765

Adjacent sequences: A068994 A068995 A068996 * A068998 A068999 A069000

Keyword

easy,nonn

Author

Benoit Cloitre, Apr 07 2002

Status

approved